Chapters 29 and 30 Game Theory and Applications

Game Theory 0 Game theory applied to economics by John Von Neuman and Oskar Morgenstern 0 Game theory allows us to analyze different social and economic situations

GAME THEORY 0 Game theory is the study of how people behave in strategic situations. 0 Strategic decisions are those in which each person, in deciding what actions to take, must consider how others might respond to that action.

Games of Strategy Defined 0 Interaction between agents can be represented by a game, when the rewards to each depends on his actions as well as those of the other player 0 A game is comprised of 0 players 0 Order to play 0 strategies 0 Chance 0 Information 0 Payoff 4

0 A Nash equilibrium is a situation in which economic actors interacting with one another each choose their best strategy given the strategies that all the others have chosen. 0 Each agent is satisfied with (i.e., does not want to change) his strategy (or action) given the strategies of all other agents. The Nash Equilibrium John Forbes Nash, Jr. June 13,

Simultaneous Games 0 Players choose at the same time and therefore do not know the choices of the other. 0 Also called imperfect information games 0 Examples: 0 Rock – paper – scissors 0 Cournot competition

Example: Coordination game Ann’ s Decision Ballet Ann gets 8 Jane gets 8 Ann gets 0 Jane gets 0 Ann gets 0 Jane gets 0 Ann gets 10 Jane gets 10 Opera Jane’s Decision Ballet Opera

Example 3: The Prisoners’ Dilemma 0 The prisoners’ dilemma provides insight into the difficulty of maintaining cooperation. 0 Often people (firms) fail to cooperate with one another even when cooperation would make them better off.

The Prisoners’ Dilemma 0 The prisoners’ dilemma is a particular “game” between two captured prisoners that illustrates why cooperation is difficult to maintain even when it is mutually beneficial.

The Prisoners’ Dilemma 0 Two people committed a crime and are being interrogated separately. 0 They are offered the following: 0 If both confessed, each spends 8 years in jail. 0 If both remained silent, each spends 1 year in jail. 0 If only one confessed, he will be set free while the other spends 20 years in jail.

Example 1: The Prisoners’ Dilemma Game Ben’ s Decision Confess Ben gets 8 years Kyle gets 8 years Ben gets 20 years Kyle goes free Ben goes free Kyle gets 20 years Ben gets 1 year Kyle gets 1 year Remain Silent Remain Silent Kyle’s Decision

Representing Games 0 Games can be represented in 0 Bi matrix/ normal form 0 Game tree/ extensive form

Toshiba IBM software game 0 Toshiba and IBM are choosing between two operating systems: UNIX or DOS 0 The two firms move at the same time 0 Imperfect information

14 Toshiba DOSUNIX IBM DOS600, , 100 UNIX100, , 600 Game of imperfect information In normal form

Game of imperfect information 15 Toshiba does not know whether IBM moved to the left or to the right, i.e., whether it is located at node 2 or node IBM Toshiba UNIX DOS UNIX DOS UNIX DOS In extensive form Information set Toshiba’s strategies: DOS UNIX

Sequential Games 0 Players move sequentially 0 A player knows all actions chosen before his move 0 Also called perfect information games 0 Examples: 0 Chess 0 Stackelberg competition

Game of perfect information 17 Player 2 (Toshiba) knows whether player 1 (IBM) moved to the left or to the right. Therefore, player 2 knows at which of two nodes it is located IBM Toshiba UNIX DOS UNIX DOS UNIX DOS In extensive form

Strategy 0 A player’s strategy is a plan of action for each of the other player’s possible actions

How to define strategies in sequential games A strategy is a plan of action for all possible outcomes/ choices made by the previous players 0 IBM: 0 Play DOS 0 Play UNIX 0 Toshiba 0 Play DOS if he plays DOS and UNIX if he plays UNIX 0 Play UNIX if he plays DOS and DOS if he plays UNIX 0 Play DOS if he plays DOS and DOS if he plays UNIX 0 Play UNIX if he plays DOS and UNIX if he plays UNIX

20 Toshiba (DOS | DOS, DOS | UNIX) (DOS | DOS, UNIX | UNIX) (UNIX | DOS, UNIX | UNIX) (UNIX | DOS, DOS | UNIX) IBM DOS600, , 100 UNIX100, , , 100 Game of perfect information In normal form Note that DOS | UNIX is read as I will play DOS if I observe him play UNIX

Equilibrium for Games Nash Equilibrium 0 Equilibrium 0 state/ outcome 0 Set of strategies 0 Players – don’t want to change behavior 0 Given - behavior of other players 0 Noncooperative games 0 No possibility of communication or binding commitments 21

Nash Equilibria 22

23 Toshiba DOSUNIX IBM DOS600, , 100 UNIX100, , 600 The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX Nash Equilibrium: Toshiba-IBM imperfect Info game

24 Toshiba (DOS | DOS, DOS | UNIX) (DOS | DOS, UNIX | UNIX) (UNIX | DOS, UNIX | UNIX) (UNIX | DOS, DOS | UNIX) IBM DOS600, , 100 UNIX100, , , Three Nash equilibria. The following outcomes satisfy the Nash criteria: 0 IBM plays DOS, Toshiba plays DOS regardless 0 IBM plays DOS, Toshiba matches IBM’s choice 0 IBM plays UNIX, Toshiba plays UNIX regardless Nash Equilibrium: Toshiba-IBM perfect Info game

Nash equilibrium: Coordinating numbers game 25 Player Player 1 11, 10, 0 2 2, 20, 0 3 3, 30, 0 4 4, 40, 0 5 5, 50, 0 6 6, 60, 0 7 7, 70, 0 8 8, 80, 0 9 9, 90, 0 100, 0 10, 10

Nash equilibrium: War game A game with no equilibria in pure strategies 26 General 2 RetreatAttack General 1Retreat5, 86, 6 Attack8, 02, 3

Nash Equilibrium: The “I Want to Be Like Mike” Game 27 Dave Wear redWear blue MichaelWear red(-1, 2)(2, -2) Wear blue(1, -1)(-2, 1) A game with no equilibria in pure strategies

Dominant Strategy Equilibria 0 Strategy A dominates strategy B if 0 A gives a higher payoff than B 0 No matter what opposing players do 0 Dominant-strategy equilibrium 0 All players play their dominant strategies 28

Oligopoly Game 29 General Motors High priceLow price Ford High price500, , 700 Low price700, , Ford has a dominant strategy to price low 0 If GM prices high, Ford is better of pricing low 0 If GM prices low, Ford is better of pricing low

Oligopoly Game 30 General Motors High priceLow price Ford High price500, , 700 Low price700, , Similarly for GM 0 The Nash equilibrium is Price low, Price low

An Equilibrium Refinement 0 Analyzing games in bi-matrix form may result in equilibria that are less satisfactory 0 These equilibria involve a non credible threat 0 The Sub Game Perfect Nash Equilibrium is a Nash equilibrium that involves credible threats only 0 It can be obtained by solving the game in extensive form using backward induction 31

32 Toshiba (DOS | DOS, DOS | UNIX) (DOS | DOS, UNIX | UNIX) (UNIX | DOS, UNIX | UNIX) (UNIX | DOS, DOS | UNIX) IBM DOS600, , 100 UNIX100, , , 100 Non credible threats: IBM-Toshiba In normal form 0 Three Nash equilibria 0 Some involve non credible threats. 0 Example IBM playing UNIX and Toshiba playing UNIX regardless: 0 Toshiba’s threat is non credible

Backward induction IBM Toshiba UNIX DOS UNIX DOS UNIX DOS

Subgame perfect Nash Equilibrium 0 Subgame perfect Nash equilibrium is 0 IBM: DOS 0 Toshiba: if DOS play DOS and if UNIX play UNIX 0 Toshiba’s threat is credible 0 In the interest of Toshiba to execute its threat

Rotten kid game 0 The kid either goes to Aunt Sophie’s house or refuses to go 0 If the kid refuses, the parent has to decide whether to punish him or relent 35 Player 2 (a parent) (punish if the kid refuses) (relent if the kid refuses) Player 1 (a difficult child) Left (go to Aunt Sophie’s House) 1, 1 Right (refuse to go to Aunt Sophie’s House) -1, -12, 0

Rotten kid game in extensive form 36 The sub game perfect Nash equilibrium is: Refuse and Relent if refuse The other Nash equilibrium, Go and Punish if refuse, relies on a non credible threat by the parent Kid Parent Refuse Go to Aunt Sophie’s House Relent if refuse Punish if refuse

Application 1: Collusive Duopoly 0 Example: The European voluntary agreement for washing machines in The agreement requires firms to eliminate from the market inefficient models 0 Ahmed and Segerson (2011) show that the agreement can raise firm profit, however, it is not stable Firm 2 eliminateKeep Firm 1 eliminate $1,000 $200$1,200 keep $1,200$200 $500

Application 2: Wal-Mart and CFL bulbs market 0 In 2006 Wal-Mart committed itself to selling 1 million CFL bulbs every year 0 This was part of Wal-Mart’s plan to become more socially responsible 0 Ahmed(2012) shows that this commitment can be an attempt to raise profit.

Wal-Mart Small firm Do not commit Commit to output target Do not Commit Do not Commit When the target is small The outcome is similar to a prisoners dilemma Application 2: Wal-Mart and CFL bulbs market

Wal-Mart Small firm Do not commit Commit to output target Do not Commit Do not Commit When the target is large When the target is large enough, we have a game of chicken Application 2: Wal-Mart and CFL bulbs market